Estimate and test parameters in a Cox proportional hazards model:

The parameter estimates for and both differ significantly from zero at the 5% level:

Compare the fitted model to the null model, containing no covariates:

Form 95% confidence intervals about relative risk and parameter estimates:

Find survival probabilities and plot model estimates:

Compare survival for different levels of with fixed at 25 with the baseline survival:

Compute the median survival time for both levels of with fixed at 25:

The baseline median survival:

The probability that an individual with levels and will survive beyond 20:

A 95% confidence interval for the same probability:

Fit a stratified model:

The number of observations in each stratum:

Plot different forms of the hazard for levels and :

The model parameter table:

Fit a model with categorical predictor variables:

Plots of the estimated CDFs for each level in the data:

Add interaction terms to the model:

No significant interaction is detected in the model:

Nominal and strata variables can be used in interaction terms:

Test model parameters against specific null hypotheses:

Test :

Test with the second parameter free:

Use a contrast matrix to test a linear combination of the parameters:

Test the following null hypothesis:

Specifying only a contrast matrix tests a hypothesis of the following form:

Show the list of available properties for a model:

Create a model with no covariates (null model):

The model is simply an estimate of the baseline survival:

An equivalent null model:

Basis functions specify which of the available variables should be included in the model:

Parameter estimates corresponding to each of the four covariates:

Not all variables must be used in the basis functions:

Basis functions also control the functional form of model terms:

Parameter estimates for the terms in the model:

Use EventData to specify right-censored or left-truncated response times:

A Cox model with randomly right-censored response:

Obtain parameter estimates and parameter confidence intervals:

The model parameters and relative risk estimates :

Confidence intervals about the parameter estimates:

Parameter summary tables:

Entries from the parameter table:

Obtain a variety of model residuals:

Martingale, Cox–Snell, and deviance residuals:

DFBETAs and scaled DFBETAS:

Score and Schoenfeld residuals:

Compute a selection of goodness-of-fit measures:

Perform global hypothesis tests:

Three tests of global model significance:

Entries from the test table:

Individual test statistics:

Test local hypotheses:

Test whether :

Entries from the test table:

Compute estimates of the SurvivalFunction , CDF , or cumulative hazard function:

The estimated SurvivalFunction for covariate levels and :

The estimated baseline SurvivalFunction , CDF , and cumulative hazard at time :

Tabulate a summary of the estimated SurvivalFunction :

A survival summary for covariate levels and :

Entries form the table:

Properties of distribution function estimates default to "SF":

Tabulate the cumulative hazard function:

Compute pointwise confidence bands and intervals:

A set of pointwise survival confidence intervals for covariate levels and :

These are "PointwiseBands" evaluated at the model "EstimationPoints":

Estimate simultaneous confidence bands and intervals:

A set of Hall–Wellner simultaneous confidence intervals for covariate levels and :

These are "HallWellnerBands" evaluated at the model "EstimationPoints":

Alternatively, use "EqualPrecisionIntervals" to measure simultaneous confidence:

Plot survival estimates and confidence bands:

Plots of baseline survival and cumulative hazard with 95% pointwise confidence bands:

The same plots for covariate levels and :

Estimate median survival times:

The median survival time for covariate levels and and at the baseline:

Estimate mean survival times:

The survival function does not reach zero due to right censoring; the mean is not well defined:

Adding a model for the remaining probability allows the mean to be estimated:

Change the confidence level for parameter confidence intervals:

Use a 90% confidence level:

The level can be changed after fitting:

Plot survival estimates with 90% confidence bands:

Plots of baseline survival with 90% pointwise confidence bands:

The level can be changed after fitting:

The setting for ConfidenceLevel is also used in summary tables:

Summarize the cumulative hazard function with 90% confidence limits:

Set the confidence range for simultaneous intervals and bands:

The bands are only defined over the confidence range:

By default the range is set to All :

Set the range to Full :

Apply transformations to confidence intervals and bands:

A collection of available transforms:

"LogLog" bands and intervals are used by default:

Transformations can ensure that intervals and bands are appropriately bounded:

Notice that linear confidence limits may exceed 1. or be less than zero:

Using a "LogLog" transform corrects this issue:

Define custom transformations:

Explicit definitions for the various named transforms:

Select the method used for handling tied event times:

The default method maximizes Breslow's partial likelihood:

When the proportion of ties is high, the estimates may differ considerably:

In the absence of tied event times, all of the methods are equivalent:

Choose an estimator for the baseline hazard:

By default, the baseline hazard estimator is matched to the tie method:

The estimators generally agree closely:

Any combination of tie and baseline method can be used:

If the last observation is censored, the survival will be Indeterminate beyond that point:

The "TailModel" suboption can control this behavior:

Some hypothetical data with two categorical predictors, species and treatment:

Multiple nominal variables can be specified in a list:

The presence of treatment appears to reduce risk significantly at the 5% level:

Create a stratified Cox model for some hypothetical data with species and treatment predictors:

Here the treatment predictor is treated as nominal and the species as a strata variable:

No parameters are estimated for strata variables:

Multiple strata can be assigned using a list:

Baseline survival functions for each of the strata:

Set the precision used during computation:

Use 30-digit precision:

A group of 16 dogs and 9 cats was selected to test a new treatment for ear mites. Each animal was randomly assigned to be treated with an older medication or the new drug. Determine whether the new treatment extends the time in weeks to ear mite reinfection:

Fit a proportional hazards model stratified on species:

The new treatment seems to significantly extend the time to ear mite reinfection:

Assess the fit using a Cox–Snell residual plot:

The plot suggests that the model is a reasonable fit to the data:

A group of 90 male patients diagnosed with cancer of the larynx was observed. Each patient was classified as having one of four stages of cancer with 1 being least severe and 4 most severe. Of interest is whether the stage of cancer is significantly linked to survival:

The table gives the relative risk of dying for each stage compared to stage 1:

The risk of dying for stage 2 patients is not different from stage 1 at the 5% level:

Find an estimate of the relative risk of death for stage 4 larynx cancer patients compared to those with stage 2 larynx cancer:

Parameter estimates for stage 2 and stage 4 larynx cancer:

Use the parameter covariance matrix to compute the standard error of the estimate:

At the 5% level, the risk of death is higher for stage 4 patients compared to stage 2 patients:

Visually compare the survival of 50-year-old patients in each stage:

Under a null hypothesis of no trend, test whether increasing the stage of larynx cancer increases the risk of dying:

Fit a Cox model stratified on stage:

LogRankTest can be used to perform a test for trend, given ordered strata:

The test statistic for trend based on the 4 levels of stage:

Reject the null hypothesis of nonpositive trend. Death rate appears to increase with stage:

A sample of 101 patients with advanced acute myelogenous leukemia was recorded. Fifty-one of these patients had received an autologous bone marrow transplant and 50 patients had an allogeneic bone marrow transplant. Of interest is whether a proportional hazards model would be valid for this data:

Fit a model stratified on transplant type:

The log cumulative hazards cross, indicating non-proportional hazards:

The difference in log cumulative hazards is not constant:

An Anderson plot of the cumulative hazards is not linear: